## H2 z h2 hy h jxy y hz h

If X, Y, and Z are principal body axes, then Eq. (17-41) reduces to

< N cc > 0--p= [Hl„-L,)hIhy+\(Ia-J^)hxh,+Z(Jxx-Jxy)hyhx]

where hx, hy, and hz are the components of the orbit normal unit vector along the principal axes.

From Eq. (17-42), we see that (1) if any principal axis is parallel to orbit normal, the secular gravity-gradient torque is zero and (2) if a principal axis is in the orbit plane, the secular gravity-gradient torque will be along that axis.

The secular gravity-gradient torque for a spin-stabilized satellite can also be calculated from Eq. (17-38). Substituting Eq. (17-35) into Eq. (17-38), the secular torque for a spinning satellite is given by

<N«v^> = —-, / *( 1 + * cos P)(Rs- X Z) dp (17-43)

Writing the unit vector Rs in terms of the true anomaly as R5=fkcosj'+qsini> and assuming that Z, p, and q are constant over one orbit, the average torque is

From Eq. (17-44), we see that (1) the secular torque is perpendicular to Z and therefore does not alter the magnitude of the angular momentum; (2) the gravity-gradient torque causes the spin axis to precess in a cone about the orbit normal with cone angle <J>=arccos(h Z); and (3) the rate of precession of Z is proportional to sin(2\$) and therefore is a maximum at <^=45 or 135 deg.

### 1722 Solar Radiation Torque

Radiation incident on a spacecraft's surface produces a force which results in a torque about the spacecraft's center of mass. The surface is subjected to radiation pressure or force per unit area equal to the vector difference between the incident and reflected momentum flux. Because the solar radiation varies as the inverse square of the distance from the Sun, the solar radiation pressure is essentially altitude independent for spacecraft in Earth orbit. The major factors determining the radiation torque on a spacecraft are (1) the intensity and spectral distribution of the incident radiation, (2) the geometry of the surface and its optical properties, and (3) the orientation of the Sun vector relative to the spacecraft.

The major sources of electromagnetic radiation pressure are (1) solar illumination (Section 5.3), (2) solar radiation reflected by the Earth and its atmosphere, i.e., the Earth's albedo (Section 4.1), and (3) radiation emitted from the Earth and its atmosphere (Section 4.2). Of these sources, as shown in Table 17-1, direct solar radiation is the dominant source and is generally the only one considered. The force produced by the solar wind is also normally negligible relative to the solar radiation pressure (see Section 5.3).

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